sph_stat_Sobolev: Finite Sobolev statistics for testing (hyper)spherical...
In sphunif: Uniformity Tests on the Circle, Sphere, and Hypersphere

sph_stat_Sobolev

R Documentation

Finite Sobolev statistics for testing (hyper)spherical uniformity

Computes the finite Sobolev statistic

S_{n, p}(\{b_{k, p}\}_{k=1}^K) = \sum_{i, j = 1}^n \sum_{k = 1}^K b_{k, p}C_k^(p / 2 - 1)(\cos^{-1}({\bf X}_i'{\bf X}_j)),

for a sequence \{b_{k, p}\}_{k = 1}^K of non-negative weights. For p = 2, the Gegenbauer polynomials are replaced by Chebyshev ones.

sph_stat_Sobolev(X, Psi_in_X = FALSE, p = 0, vk2 = c(0, 0, 1))

cir_stat_Sobolev(Theta, Psi_in_Theta = FALSE, vk2 = c(0, 0, 1))

`X`	an array of size `c(n, p, M)` containing the Cartesian coordinates of `M` samples of size `n` of directions on `S^{p-1}`. Must not contain `NA`'s.
`Psi_in_X`	does `X` contain the shortest angles matrix `\boldsymbol\Psi` that is obtained with `Psi_mat(X)`? If `FALSE` (default), `\boldsymbol\Psi` is computed internally.
`p`	integer giving the dimension of the ambient space `R^p` that contains `S^{p-1}`.
`vk2`	weights for the finite Sobolev test. A non-negative vector or matrix. Defaults to `c(0, 0, 1)`.
`Theta`	a matrix of size `c(n, M)` with `M` samples of size `n` of circular data on `[0, 2\pi)`. Must not contain `NA`'s.
`Psi_in_Theta`	does `Theta` contain the shortest angles matrix `\boldsymbol\Psi` that is obtained with `Psi_mat(array(Theta, dim = c(n, 1, M)))`? If `FALSE` (default), `\boldsymbol\Psi` is computed internally.